Question

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, projvu u = , v =

Answer 1

The vector u represents the sum of two orthogonal vectors, and which can be written as,

$u=Proj_vu+(u-Proj_vu)$.

To find the answer, we have to know about the projection of vectors.

How to write u as a sum of two orthogonal vectors?

• Let u and v are two vectors, then the projection of the vector u onto the vector v is given by

                               $Proj_v u=\frac{u.v}{v.v} v$  (inner product)

• let  

              $u=(u_1,u_2)\\v=(v_1,v_2)$    then, projection of u on v is,

                              $Proj_vu=\frac{u_1v_1+u_2v_2}{v_1v_1+v_2v_2} (v_1,v_2)$

• Then, u can be written as the sum of two orthogonal vectors as,

                 $u=Proj_vu+(u-Proj_vu)$

Thus, we can conclude that, the vector u represents the sum of two

orthogonal vectors, and which can be written as,

$u=Proj_vu+(u-Proj_vu)$.

Learn more about orthogonal vectors here:

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